Internet can be delivered over Ethernet or coax.
If you have TV service, the ONT is usually configured to use coax. So, yes, you would need to have the router connected to a coax network that is connected to the ONT.
If you ordered internet only service, it's possible the ONT is configured for Ethernet. You can test this by plugging an Ethernet cable into the ONT's network jack and the router's WAN Ethernet port. If it works, you're in business.
Verizon can tell you how your ONT is provisioned if you contact them (note this is a peer-to-peer forum).
As to why the coax is disconnected? I'm afraid no one here can answer that. A Verizon tech doesn't need to visit when service is shut down, so there is no way to know who removed the coax. For that matter, we don't even know if it was connected before. Better to worry about getting it working now than how it got that way.
You'll probably get useful help if you contact Verizon's support team.
Good luck.